Related papers: On the Hardness of the One-Sided Code Sparsifier P…
We consider the problem of estimating the spectral density of the normalized adjacency matrix of an $n$-node undirected graph. We provide a randomized algorithm that, with $O(n\epsilon^{-2})$ queries to a degree and neighbor oracle and in…
The Planar Graph Metric Compression Problem is to compactly encode the distances among $k$ nodes in a planar graph of size $n$. Two na\"ive solutions are to store the graph using $O(n)$ bits, or to explicitly store the distance matrix with…
Discrepancy theory provides powerful tools for producing higher-quality objects which "beat the union bound" in fundamental settings throughout combinatorics and computer science. However, this quality has often come at the price of more…
The notion of vertex sparsification is introduced in \cite{M}, where it was shown that for any graph $G = (V, E)$ and a subset of $k$ terminals $K \subset V$, there is a polynomial time algorithm to construct a graph $H = (K, E_H)$ on just…
We show that it is quasi-NP-hard to color 2-colorable 8-uniform hypergraphs with $2^{(\log N)^{1/10-o(1)}}$ colors, where $N$ is the number of vertices. There has been much focus on hardness of hypergraph coloring recently. Guruswami,…
Graph compression or sparsification is a basic information-theoretic and computational question. A major open problem in this research area is whether $(1+\epsilon)$-approximate cut-preserving vertex sparsifiers with size close to the…
We continue the investigation of polynomial-time sparsification for NP-complete Boolean Constraint Satisfaction Problems (CSPs). The goal in sparsification is to reduce the number of constraints in a problem instance without changing the…
Multiplicative cut sparsifiers, introduced by Bencz\'ur and Karger [STOC'96], have proved extremely influential and found various applications. Precise characterisations were established for sparsifiability of graphs with other 2-variable…
A hypergraph spectral sparsifier of a hypergraph $G$ is a weighted subgraph $H$ that approximates the Laplacian of $G$ to a specified precision. Recent work has shown that similar to ordinary graphs, there exist $\widetilde{O}(n)$-size…
We provide the first online algorithm for spectral hypergraph sparsification. In the online setting, hyperedges with positive weights are arriving in a stream, and upon the arrival of each hyperedge, we must irrevocably decide whether or…
We present a nearly-linear time algorithm that produces high-quality sparsifiers of weighted graphs. Given as input a weighted graph $G=(V,E,w)$ and a parameter $\epsilon>0$, we produce a weighted subgraph $H=(V,\tilde{E},\tilde{w})$ of $G$…
In this letter, we propose a sparsification method for precoding codebooks that reduces the peak-to-average power ratio (PAPR) while preserving the achievable rate. By exploiting the fact that precoder matrices lie on the Grassmann…
We give a 2-approximation algorithm for Non-Uniform Sparsest Cut that runs in time $n^{O(k)}$, where $k$ is the treewidth of the graph. This improves on the previous $2^{2^k}$-approximation in time $\poly(n) 2^{O(k)}$ due to Chlamt\'a\v{c}…
The problem of CSP sparsification asks: for a given CSP instance, what is the sparsest possible reweighting such that for every possible assignment to the instance, the number of satisfied constraints is preserved up to a factor of $1 \pm…
The seminal work of Bencz\'ur and Karger demonstrated cut sparsifiers of near-linear size. Subsequent extensions have yielded sparsifiers for hypergraph cuts and more recently linear codes over Abelian groups. A decade ago, Kogan and…
Uncertain graphs are prevalent in several applications including communications systems, biological databases and social networks. The ever increasing size of the underlying data renders both graph storage and query processing extremely…
Graph sparsification underlies a large number of algorithms, ranging from approximation algorithms for cut problems to solvers for linear systems in the graph Laplacian. In its strongest form, "spectral sparsification" reduces the number of…
Pioneered by Benczur and Karger for cuts in graphs [STOC'96], sparsification is a fundamental topic with wide-ranging applications that has been studied, e.g., for graphs and hypergraphs, in a combinatorial and a spectral setting, and with…
The (non-uniform) sparsest cut problem is the following graph-partitioning problem: given a "supply" graph, and demands on pairs of vertices, delete some subset of supply edges to minimize the ratio of the supply edges cut to the total…
Stabilizer codes are the most widely studied class of quantum error-correcting codes and form the basis of most proposals for a fault-tolerant quantum computer. A stabilizer code is defined by a set of parity-check operators, which are…